−Table of Contents
Complemented lattices
Abbreviation: CdLat
Definition
A \emph{complemented lattice} is a bounded lattices L=⟨L,∨,0,∧,1⟩ such that
every element has a complement: ∃y(x∨y=1 and x∧y=0)
Morphisms
Let L and M be complemented lattices. A morphism from L to M is a function h:L→M that is a bounded lattice homomorphism:
h(x∨y)=h(x)∨h(y), h(x∧y)=h(x)∧h(y), h(0)=0, h(1)=1
Examples
Example 1: ⟨P(S),∪,∅,∩,S⟩, the collection of subsets of a set S, with union, empty set, intersection, and the whole set S.
Basic results
Properties
| Classtype | first-order |
|---|---|
| Equational theory | decidable |
| Quasiequational theory | |
| First-order theory | undecidable |
| Locally finite | no |
| Residual size | unbounded |
| Congruence distributive | yes |
| Congruence modular | yes |
| Congruence n-permutable | yes |
| Congruence regular | no |
| Congruence uniform | no |
| Congruence extension property | no |
| Definable principal congruences | no |
| Equationally def. pr. cong. | no |
| Amalgamation property | |
| Strong amalgamation property | |
| Epimorphisms are surjective |
Finite members
f(1)=1f(2)=1f(3)=0f(4)=1f(5)=2f(6)=f(7)=f(8)=
